MathDB

156

Part of 1971 All Soviet Union Mathematical Olympiad

Problems(1)

ASU 156 All Soviet Union MO 1971 n^3 cubes inside a cube of length n

Source:

7/3/2019
A cube with the edge of length nn is divided onto n3n^3 unit ones. Let us choose some of them and draw three lines parallel to the edges through their centres. What is the least possible number of the chosen small cubes necessary to make those lines cross all the smaller cubes?
a) Find the answer for the small nn (n=2,3,4n = 2,3,4).
b) Try to find the answer for n=10n = 10.
c) If You can not solve the general problem, try to estimate that value from the upper and lower side.
d) Note, that You can reformulate the problem in such a way:
Consider all the triples (x1,x2,x3)(x_1,x_2,x_3), where xix_i can be one of the integers 1,2,...,n1,2,...,n. What is the minimal number of the triples necessary to provide the property:
for each of the triples there exist the chosen one, that differs only in one coordinate.
Try to find the answer for the situation with more than three coordinates, for example, with four.
combinatorial geometrycube